Ended up having to repeat the process for the right mouse button. This time I managed to knock off one of the solder pads. Eventually by applying a lot of pressure and heat though, I was able to get the switch to bind with whatever was left of it, although I still have no idea how long my fix will actually hold for. But, something to watch out for if you're doing this.

Got this mouse, had this problem, sent it off for an RMA. The one they sent back eventually developed this exact same issue. Here's to hoping I won't have to do this again for a while now. Thanks for the guide.